Chapter 7: Problem 1
Write down the Lagrangian for a projectile (subject to no air resistance) in terms of its Cartesian coordinates \((x, y, z),\) with \(z\) measured vertically upward. Find the three Lagrange equations and show that they are exactly what you would expect for the equations of motion.
Short Answer
Expert verified
The Lagrangian is \( L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - mgz \). The equations: \( \ddot{x} = 0 \), \( \ddot{y} = 0 \), \( \ddot{z} = -g \).
Step by step solution
01
Understanding the Lagrangian Function
The Lagrangian function, defined as \( L = T - V \), where \( T \) is the kinetic energy and \( V \) is the potential energy, is used to derive the equations of motion. For a projectile in three-dimensional space, we'll express both the kinetic and potential energy in terms of the Cartesian coordinates \((x, y, z)\).
02
Express Kinetic Energy in Cartesian Coordinates
The kinetic energy \( T \) of a projectile with mass \( m \) moving with velocity components \( \dot{x}, \dot{y}, \dot{z} \) is given by \( T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) \). These are the squared velocity components in the Cartesian system.
03
Express Potential Energy in Cartesian Coordinates
The potential energy \( V \) of a projectile in a gravitational field (with constant gravity \( g \)) depends on the vertical position \( z \). It is expressed as \( V = mgz \), where \( z \) is the height above the reference point.
04
Write the Lagrangian
Substitute the expressions for kinetic and potential energy into the Lagrangian: \[ L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - mgz \] This function represents the Lagrangian of the system.
05
Derive the Lagrange Equations
Using the Euler-Lagrange equation, \( \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 \), apply it to each coordinate \( x, y, \) and \( z \).
06
Equation for the x-direction
Applying the Lagrange equation to \( x \):\[ \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}} \right) = m\ddot{x} \quad \text{and} \quad \frac{\partial L}{\partial x} = 0 \]Thus, the equation becomes: \[ m\ddot{x} = 0 \] indicating no acceleration in the \( x \)-direction.
07
Equation for the y-direction
Similarly, apply the Lagrange equation to \( y \):\[ \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{y}} \right) = m\ddot{y} \quad \text{and} \quad \frac{\partial L}{\partial y} = 0 \]Thus, \[ m\ddot{y} = 0 \] indicating no acceleration in the \( y \)-direction as well.
08
Equation for the z-direction
For the \( z \) coordinate, we have:\[ \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{z}} \right) = m\ddot{z} \quad \text{and} \quad \frac{\partial L}{\partial z} = -mg \]This results in:\[ m\ddot{z} = -mg \] which represents the acceleration due to gravity.
09
Summary of Equations of Motion
The Lagrange equations lead to the expected equations of motion: \[ \ddot{x} = 0, \quad \ddot{y} = 0, \quad \ddot{z} = -g \] These equations show that the projectile moves with constant velocity in the \( x \) and \( y \) directions, and with constant acceleration \( -g \) in the \( z \) direction.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Projectile Motion
Projectile motion describes the path of an object thrown into space, subject to gravitational forces alone. In our exercise, we're analyzing this motion without air resistance, focusing on three-dimensional space with coordinates
- \( (x, y, z) \)
- Initial Velocity: Determines how the object starts its trajectory.
- Gravity: Acts downward, affecting vertical motion.
Euler-Lagrange Equation
The Euler-Lagrange equation is a fundamental principle to derive equations of motion in Lagrangian mechanics. It connects the Lagrangian function to physical trajectories of systems. The equation is:\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0\]Where \( L = T - V \), it represents the difference between kinetic and potential energy. Here’s how it applies:
- For each coordinate (\( x, y, z \)): We take partial derivatives related to velocity and position.
- Motion Derivation: This leads to equations describing how the system evolves over time.
Kinetic and Potential Energy
Kinetic and potential energies are key to understanding Lagrangian mechanics. They are components of the Lagrangian function \( L = T - V \), representing how energy changes in motion.
Kinetic Energy (T)
Kinetic energy is associated with the motion of the projectile. Given by:\[ T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) \]- Mass \( m \): How heavy the object is.
- Velocity Components (\( \dot{x}, \dot{y}, \dot{z} \)): Speed in the three axes.
Potential Energy (V)
Potential energy relates to the object's position in a gravitational field:\[ V = mgz\]- Gravitational Force (\( g \)): Constant pulling the object down.
- Height (\( z \)): Position elevated above a reference point.